Place the white chocolate pieces and the half and half into a microwave safe bowl. Heat for 1 minutes, stopping to stir half way through. Continue to heat until melted and smooth. Add the cocoa and sugar, stirring constantly until thoroughly blended. Remove from heat and slowly add 1/2 cup of the milk and stir until smooth. Cool to room temperature.

In a blender place the remaining cup of milk, the room temperature chocolate mixture, and the ice. Blend on high speed until smooth and the consistency of a frozen daiquiri. Pour into a cup and top with homemade whipped cream and chocolate shavings.